Internat. J. Math. & Math. Sci. 451 VOL. 18 NO. 3 (1995) 451-462
COEFFICIENT SUBRINGS OF CERTAIN LOCAL RINGS WITH PRIME-POWER CHARACTERISTIC
TAKAO SUMIYAMA Department of Mathematics Aichi Institute of Technology Yakusa-ch6, Toyota 470-03 Japan
(Received October 13, 1993 and in revised form September 15, 1994)
ABSTRACT. If R is a local ring whose radical J(R) is nilpotent and R/J(R) is a commutative field which is algebraic over GF(p), then R has at least one subring S such that S w*=,S,, where each S, is isomorphic to a Galois ring and S/J (S) is naturally isomorphic to R/J (R). Such subrings ofR are mutually isomorphic, but not necessarily conjugate in R. KEY WORDS AND PHRASES: Coefficient ring, Galois ring, local ring, Szele matrix. 1991 AMS SUBJECT CLASSIFICATION CODES: 16L30, 16P10, 16D70
0. INTRODUCTION Let p be a fixed prime. For any positive integers n and r, there exists up to isomorphism a unique r-dimensional separable extension GR(p",r) of Z/p"Z, which is called the Galois ring of characteristic p" and rank r (see [9, p. 293, Theorem XV.2]). This ring was first noticed by Krull [8], and was later rediscovered by Janusz [6] and Raghavendran [12]. By Wedderburn-Malcev theorem (see, for instance, [4, p. 491 ]), if R is a finite dimensional algebra over a field K such that R R/J(R) is a separable algebra over K, then R contains a semisimple subalgebra S such that R S J(R) (direct sum as vector spaces). Such subalgebras of R are conjugate each other. Concerning the case R is not an algebra over a field, Raghavendran 12, Theorem 8], Clark [3] and Wilson 17, Lemma 2.1 have proved the following: If R is a finite local ring with characteristic p" whose residue field is GF(p r), then R contains a subring S such that S is isomorphic to GR(p",r) (hence R S + J (R)). Such a subring S ofR is called a coefficient ring of R. Coefficient rings of R are conjugate each other. We can embed R to a ring of Szele matrices over S (see 1). If R is a finite ring of characteristic p", then R contains a subring T (unique up to isomorphism) such that (1) R T N (as abelian groups), where N is an additive subgroup of J(R), (2) T is a direct sum of matrix rings over Galois rings, (3) J(T) TnJ(R) pT, and (4) R T + J(R). The purpose of this paper is to extend these results to certain rings which are not necessarily finite. 1. In what follows, when S is a set, SI will denote the cardinal number of S. When A is a ring, for any subset S of A, (S) denotes the subring of A generated by S. A ring A is called locally finite if any finite subset of A generates a finite subring. When A is a ring with 1, for B to be called a subring of A, B must contain 1. Let J(A) denote the Jacobson radical of A, Aut(A) the automorphism group of A, and (A),, the ring of n n matrices having entries in A. If A 9 1, A* denotes the group of units of A. For a e A *, o (a) denotes the multiplicative order of a. The Galois ring GR(p",r) is characterized as a ring isomorphic to (Z/p"Z) [X]/(f(X) (Z/p"Z) [X]), where f(X) (Z/p"Z)[X] is a monic polynomial of degree r, and is irreducible modulo p Z/p"Z (see [7, Chapter XVI]). By [12, Proposition 1], any subring of GR(p",r) is isomorphic to GR(p",s), where 452 T. SUMIYAMA
s is a divisor of r. Conversely, if s is a divisor of r, then there is a unique subring of GR(p",r) which is isomorphic to GR (p", s ). The following lemma is easily deduced from 16, Theorem 3 (I)] and its proof.
LEMMA 1.1. Let R be a finite local ring of characteristic p" whose residue field is GF(pr). If a e R* satisfies o (a) p r- 1, then the subring (a) of R is isomorphic to GR (p", r).
A ring R will be called an IG-ring if there exists a sequence {R,}7= of subrings of R such that R, R,. ,, R, GR(p",r,) (i > 1) and R '= R,, where {r,}= is a sequence of positive integers such that r, r, l(i > 1). If R is an IG-ring described above, then R, is the only subring of R which is isomorphic to GR(p",r,). So we can write R =wT'=GR(p",r,). Let p be a prime, n a positive integer and r < r < an infinite sequence of positive integers such that r, r, . By the fact we observed above, there exists a natural embedding t',+'GR(p",r,)---GR(p",r,/)foreachi> 1. Letusputt,=idaR(p,,.r, landt,=r:_ t_2o...ot,j-! I+l for
PROOF. (I) and (II). For each > 1,
0 --) pGR(p",rl) GR(p",r,) GR(p,r,) --) 0 is an exact sequence of GR(p",r,)-modules. So- we get the result-- by [2, Chapitre 2, 6, n 6, Proposition 8]. (III) Let us put K 7'= GF(p '). Let W,(K) be the ring of Witt vectors over K of length n (see [15, Chapter II, 6] or [5, Kapitel II, 10.4]. By (I) and [5, Kapitel II, 10.4], both R and W,(K) are elementary complete local rings (in [14], elementare vollstandige lokale Ringe)of charac- teristic p" whose residue fields are K. Since an elementary complete local ring is uniquely determined by its characteristic and residue field (see [14, Anhang 2]), we see that R is isomorphic to W,(K). Let W(K) be the ring of Witt vectors over K of infinite length. By [7, Chapter V, 7], W(K) is a discrete COEFFICIENT SUBRINGS OF CERTAIN LOCAL RINGS 453
valuation ring whose radical s generated by p. Since W(K) s the projective limit of {W,,(K)}'= (see [15, Chapter II, 6]), W,,(K) is a homomorphic image of W(K). (IV) IfRisadiscretevaluatlonringwithradicalpR,thenanyidealofRisoftheformpJR(] >0), so the result is clear from (III). (V) Clear from (III), since any proper homomorphic image of a principal ideal domain is self-injectlve. (VI) Immediate by [9, p. 294, Corollary XV.3].
Let {rl}*__ be an infinite sequence of positive integers such that r and rlr/(l > 1), and S 7'-_ GR(p", r) be an IG-ring of characteristic p". Let n n > n2 > > n, be a nonincreasing sequence of positive integers. Let us put Sj wT--t GR(p ''j, r) for < j < t. Let S ---> S be the natural homomorphism followed by the isomorphism S/p"JS--S of Proposition"1.2 (II). Let us put U(S;n,,n2,.. nt) {(,j)E(S) Ic,jEp' ifi>j} It is easy to see that U(S;n,n2,.. n,)forms a subring of (S)t,. Let M(S;n,n rtt) denote the set of all t matrices (a,), where a, Sj, and a, p'-"S for > j. Let @ be the mapping of U(S;n,nz n,) onto M(S;n,n n,) defined by (,) t- (a,) where a,j (c,). It is easy to check that addition and multiplication in M(S;nt,n2 n,) can be defined by stipulating that @ preserves addition and multiplication. Following [17], we call M(S ;n,nz n) a ring of Szele matrices over S.
LEMMA 1.3. (cf. 17, Lemma 2.1 ]) Let R be a ring with which contains an IG-subring S of characteristic p". If R is finitely generated as a left S-module, then there exists a nonincreasing sequence n n > nz > > nt of positive integers such that R is isomorphic to a subring of M(S;nl,n nt).
PROOF. By Proposition 1.2 (V), there exists a submodule N of R such that R S @ N as left S-modules. By Proposition 1.2 (III), there are a discrete valuation ring W and a homomorphism of W onto S. By defining ay=(a)y (a W, y6N), N is a finitely generated W-module. Since a finitely generated module over a principal ideal domain is a direct sum of cyclic modules, there exist Yt, Y y E N such that N @= Wy,. Let s + 1, x and x, y,_ (2 < < t). Then we get R @= Sx,. Let Sx, S/p"'S as S-modules (n n). Without loss of generality, we may assume nl > n2 > > n,. For each a R, we can write _ S) x,a Y ,x (, Since O=p x,a ,7: p ct, x by Proposition 1.2 (IV), x, p' -rttS if > j. As , is uniquely determined modulo p'S by a, we can define :R ---) M(S;n,n n) by a I-) ((o,)). It is easy to see that 1/is an injective ring homo- morphism. 2.
Let G be a group, and N a normal subgroup of G. Let p :G --) H G/N be the natural homo- morphism. A monomorphism k H G will be called a right inverse of 9 if 9 k idn. If k is a right inverse of p, then G is a semidirect product of N and .(H). The following lemma is a variation-- of Schur-Zassenhaus theorem 13, Chapter 9, 9.1.2]. 454 T. SUMIYAMA
LEMMA 2.1. Let G be a group, and N a normal subgroup of G. Let p" G -4 H G/N be the natural homomorphism. Assume that N s locally finite, and there exists a sequence {H,}7'= of finite subgroups of Hsuch that H, cH,+(i > I).v.)7":H,=H and. for any a e N and any > l,o(a) and [H,[ are copnme. Then: (I) There exists a right nverse 'H G of p. (II) If, for some m > 1, there exists a monomorphism- lu" H,,, --> G such that 9 It' id.,,,, then there exists a right inverse It" H --4 G of 9 such that It ],,,= It'. (III) There exists a unique right inverse of 9 if and only f G is a ndpotent group. (IV) If It' H,,, ---9 G and la"" H,,, -9 G are monomorphsms such that 9 It' 9 It" id.,,,, then It'(H,,,) and It"(H,,,) are conjugate n G.
PROOF. (I) For each x e H, we can choose an element g, of G such that 9(g)=x. As G is locally fimte (see [13, Chapter 14, 14.3.1]), the subgroup G of G generated by {g,}., is finite, and p 1, is a homomorphism of G onto H,. Let us put N =Ker(9 ],). Since [N[ and ]H[ are coprime, by Schur-Zassenhaus theorem 13, Chapter 9, 9.1.2], there exists a right inverse . Ht ----) Gi of 9 Ic,, Next, let {g,'}, . be a set of elements of G such that 9(g,') y for any y H, and finite Then is a {g} n, {g,'},, ," Let G2 be the subgroup of G generated by {g,'}, ,... 9 1: homomorphism of G2 onto H2. By [13, Chapter 9, 9.1.3], there exists a complement subgroup L of N=Ker(p I) in G2 such that L ,(H). The mapping Lz'Hz---)Gz defined by H G/N bN_ b(b L) is a right inverse of 9 12- For any a H L(a)-.l(a) NoL { }, hence we see L In,= k. Continuing this process inductively, we get a sequence G G2 c of finite subgroups of G and a sequence of right inverses k," H, G, of such that ., for any {k,}'__ 9 1, . In, _< < j. Then . lim_ %, H wT__ H, G is a right inverse of 9. (II) can also be proved in the same way by starting from - H,,, It'(H,). -- It" (III) Assume that ." H G is the unique right inverse of p. Then-- G is a semidirect product of N and ,(H). We shall show that this is the direct product. Suppose that there exist c e N and z e H such that c.(z) .(z)c. Let us define-- --) G by It(b) z-k(b)z. Then is a right inverse of p different It'H It from ,, which contradicts our hypothesis. So G is the direct product ofNand .(H). Hence G is nilpotent. Conversely, let us suppose that G is nilpotent, and . and Ix are right inverses of 19. For each > 1, let G, be the subgroup of G generated by k(H,)w It(H,). Then 9 1, is a homomorphism of G, onto H,. Both ,(H,) and It(H,) are complement subgroups for N, Ker(p I,) in G,. Since G, is a finite nilpotent group, for each prime divisor q of G,I, G, contains a unique q-Sylow subgroup. Each G, is the direct product of such Sylow subgroups. As n, and N, are coprime, we have X(H,) It(H,). So , In,= It In,. Since this holds for each > 1, we see . It. (IV) Let L be the finite subgroup of G generated by It'(H) t.) It"(H,). Then 9 It. is a homomorphism of L onto Hm. Since lKer(9 Iz)l IN LI and n,,I are coprime, by Schur-Zassenhaus theorem, It'(H,) and It"(H,,,) are conjugate in L.
Let G, N, H and p" G H be as in Lemma 2.1. We say that G has property (GC) with respect to N if, for any two right inverses It and v of 9, It(H) and v(H) are conjugate in G. If H is finite, then by Lemma 2.1 (IV), G has the property (GC) with respect to N. COEFFICIENT SUBRINGS OF CERTAIN LOCAL RINGS 455
Let R be a ring wth 1. Let S be a subring of R, and I J(R)S. The homomorphism of S/I to R/J(R) defined by a + I a +J(R)(a e S) is in.lective. We shall say that S/I is naturally isomorphic to R/J(R) if this homomorphism is onto. If S is a local subring of a local ring R and if J(S) is nilpotent, then J(S)= J(R)S. Now we shall state main theorems of this section, which generalize the result of R. Raghavendran [9, p. 373, Theorem XIX.4].
THEOREM 2.2. Let R be a local ring with radical M. Assume that M is nilpotent, and K R/M is a commutative field of characteristic p (p a prime) which is algebraic over GF(p). Then there exists an IG-subring S of R such that S/pS is naturally isomorphic to K.
PROOF. Since K is algebraic over GF(p), KI is either finite or countably infinite. So there exists a sequence {K,}*= of finite subfields of K such that K, c K, ,(i > 1) and 7_-, K, K. Let K, GR(p r,). The natural homomorphism rt R K induces a group homomorphism x* : [R. of R* onto K*. Each (1 +M')/(I +M'*) is isomorphic to the additive group M'/M '+l. As pM' M'/, the order of each element of + M Kern* is a power-- ofp. Furthermore, K* 7'= K,*, where ]K,*[ p is coprime to p. So, by Lemma 2.1 (I), there exists a right inverse : K* --) R* of rt*. For each > 1, let z, be a generator of K,*. By Lemma 1.1, the subring S, (X(ot,)) of R is isomorphic to GR(p",r,), where p" is the characteristic of R. Consequently, S (k(K*)) 7=S, is an IG-subring of R, and SIpS is naturally isomorphic to K.
Such a subring S of R stated in Theorem 2.2 will be called a coefficient subring of R. When R is a commutative local ring satisfying the assumption of Theorem 2.2, S coincides with the subring described in 11, p. 106, Theorem 31.1 ]. Let R, M, S and K t3*= GF(p ') be as in Theorem 2.2, where {r,}7= is a sequence of positive integers such that r, r, t(i > 1). Let p" be the characteristic of R. Let S" be another coefficient subring of R. From what was stated in 1, S' 7'-- GR (p", r,), which is isomorphic to S. By Proposition 1.2 (V), there exists a left S'-submodule N of R such that R S" N as left S'-modules. Ifk K* --> R * is a right inverse of x*, then by the proof of Theorem 2.3, S Q,,(K*)) is a coefficient subring of R. We shall show that, if , and It are different right inverses of n:*, then (L(K*)) (It(K*)). Let us suppose ((K*)) (It(K*)) and denote it by S. Let {K,}7'__ be a sequence of finite subfields of K such that K, GF(p '), K, K, t(i > 1) and t37"= K, K. As X, : It, there exist a number j > and an element of K such that k(cz) :g: It(o:). By Lemma 1.1, both T (L(K*)) and T'= (l.t(K*)) are isomorphic to GR (p", r). As S wT-- (,(K,*)), there exists a number > such that T t3 T' c:: 0,,(K*)). Since QL(Kt*)) is a Galois ring, T _-- T' implies T T'. The restriction x It. is a homomorphism of T* onto K,*. Both ),, I-. and It Ix,. are right inverses of rt: It., so T* is the direct product of L(K*) and Ker(x It.) + pT, and is also the direct product of It(K*) and +pT. As IK*I and [1 +pTI are coprime, we have X(K*) It(K*). So there exists some 13 K* such that () It(J3). Then cz n:* ),,(cz) n:* it(l)= 13, which means ),.(o:) la(x). This contradicts our choice of By making use of Lemma 2.1 (I), we can easily see that, if S is a coefficient subring of R, there exists a right inverse ),.: K* --) S* of x* such that S Q,.(K*)). Summarizing the above, we obtain the following theorem.
THEOREM 2.3. Let R be a local ring with radical M. Assume that M is nilpotent, and K R/M is a commutative field of characteristic p (p a prime) which is algebraic over GR (p). Let n* R* ---) K* 456 T. SUMIYAM
be the group homomorphism induced by the natural ring homomorphism :" R K. Then" (I) If S' is a coefficient subrlng of R, then there exists a S'-submodule N of R such that R S' @ N as left S'-modules. (II) All coefficient subnngs of R are isomorphic. (III) If 2,." K* R* is a right inverse of n:*, then S ()(K*)) is a coefficient subring of R. Con- versely, if S is a coefficient-- subnng of R, then there exists uniquely a right inverse such that S ((K*)). (IV) All coefficient subrings of R are conjugate in R if and only if R* has property (GC) with respect to + M.
With the same notation as in Theorem 2.2, M/M is regarded as a left K-space by the operation
a x ax (-d K R/M,- e M/M2).
THEOREM 2.4. Let R be a local ring with radical M. Assume that M is nilpotent, and K R/M is a commutative field of characteristic p (p a prime) which is algebraic over GF(p). Let Sbe a coefficient subring of R. Then R is finitely generated as a left S-module if and only if M/M is a finite dimensional left K-space. In this case, there exists a finitely generated left S-submodule N of M such that R S N as left S-modules, and there exists a nonincreasing sequence nl > ii _> _> n, of positive integers (p"' is the characteristic of R) such that R is isomorphic to a subring of M (S ;n,n2 n,).
PROOF. Assume that R is finitely generated as left S-module. Then R is a Noetherian left S-module, since S is a Noetherian ring by Proposition 1.2 (IV). As M is a left S-submodule of R, M is a finitely generated left S-module. This implies that M/M is a finite dimensional left K-space. Conversely, let us assume that M/M is a finite dimensional left K-space. Let t.o be the nilpotency index of M. Let x,x:, xa be elements of M whose images modulo M form a K-basis of M/M 2. As S/pS is naturally isomorphic to K, any element of y of M is written as y=a,=a,x,+ y" (a, S,y' M2). Let z XJ:, b,x + z' (b S,z' b be another element of M. Then yz=Y.,=,a,x,bjxj+w'" (w'" e M3). Each x,b is written as x,b Z= c, xa + w,j" (q,j e S, w,' e M2). So we see that any element v' of M can be written as v'='[4=a,x,x+v'" (a, S,v" M3). Continuing in this way, we see that any element of M is written as S-coefficient line combination of distinct products of or fewer x,'s. So M is a finitely generated left S-module. Also K R/M is a finitely generated left S-module, hence R is a finitely generated left S-module. Now suppose that R-is finitely generated as left S-module. By Theorem 2.3 (I), there exists a finitely generated left S-submodule N' of R such that R S N' as left S-modules. By Proposition 1.2 (III), there exist a discrete valuation ring V and a homomohism of V onto S. Defining ay (a )y (a V, y N'), we can regard N' as a left V-module. Then there exist x,x2 xt N" COEFFICIENT SUBRINGS OF CERTAIN LOCAL RINGS 457
such that N'= @i= Vx, )I= ,Sx,. By putting x0 1, we get R @_-oSx,. Let c,,c2 c, be elements of S such that c, x, under the natural homomorphism :" R K. Let us put Y0 and y, x,- c, for
Let R be a local ring described in Theorem 2.2. Then R may have more than one coefficient subring. Concerning this subject, first we can state the following.
THEOREM 3.1. Let Tbe an IG-ring of characteristic p" different from GR(p", 1). Then, for any infinite cardinal number %, there exists a local ring R such that (1) M J(R) is nilpotent, (2) K R/M is a commutative field of characteristic p (p a prime) which is algebraic over GF(p), (3) coefficient subrings of R are isomorphic to T, (4) all coefficient subrings of R are conjugate in R, and (5) % is the number of all coefficient subrings of R.
PROOF. Let T=w7=GR(p",r,), where {r,}*= is a sequence of positive integers such that r, r, (i > 1). Let K T/pT and T K be the natural homomorphism. As K is a proper extension of GF(p), there exists an automorphism" of K different from idK. Let be the automorphism of T which induces t modulo pT (see Proposition-- 1.2 (VI)). LetA be a set of cardinality %, and V , ,T be a free T-module. The abelian group T V together with the multiplication (a,x) (a',x') (aa',ax" + (a')x) forms a ring, which we denote by R. Let n:'R K be the homomorphism defined by (a,x) t- '(a), and M Kerry. As R/M K and M l= O, R is a local ring with radical M whose residue field is K. By Theorem 2.3 (1117, there exists a one-to-one correspondence between the set of all coefficient subrings of R and the set Yof all right inverses of :* = IRo" R* K*. By the embedding T 9 a (a, 0) R, T is regarded as a coefficient subring of R. So, by Theorem there exists a inverse ." K* R* of :* such-- that T. Since K GF(pr'), 2.3 (liD, right (,(K*)) wT--t there exists a number j > such- that is not the identity on GF(p ). Let a generator of GF(p ')*, 7be and c ,(?). It is easy to see that, for any z V, R* h (c, z) is of multiplicative order p 1. So, for each z V, we can define a group homomorphism tz" GF(pr) * R* by (c,z)'. By Lemma 2.1 (II), we can extend t.tz" to l.t Y. If V z,z2 and Z :g: Z2, then : ll, So YI > VI %. l.tt z2. - Let S be a coefficient subring of R. We shall show that S is conjugate to T. By Theorem 2.3 (III), there exists a right inverse ," K* R* of n:* such that S (,'(K*)). Let ,'(T) (c',z), where c" T and z V. Let U be the finite subgroup of R* generated by .(?) and .'(T). As the restriction n: Iv is a homomorphism of U onto -GF(p)*, by Schur-Zassenhaus theorem, there exists (b, w) R* (b T, w e V) and an integer such that ,'('f)=(b,w)-.(')(b,w). Then, (c',z)=(b,w)-i(c',O)(b,w), which implies c'=c'. As rg(c')=lt(/(c'))=?=(,(?))='(c), so c'=c and /(?) (c,z). Let x {c-(c)}-lz. Suppose that cte K satisfies ct =?for some integer m. Let ,(ct)= a. Then, by the same reason as above, we can write .'(ct)= (a, y) for some y e V. 458 T. SUMIYAMA
As (c, Z '(') k'(O") (a, y)'" (a'", a" (o(a))'" a or(a }-'y ), we get c =a" and z {c -o(c)} {a -o(a)}-'y. So (l,x)t'(ot)=(a,y +o(a)x)=(a,ax)=l(a)(l,x). As K* is the union of cyclic subgroups generated by such o which contain GF(p")* (generated by this prove.,, S (U(K*))= (1,x)-'T(1,x). So YI, the number of all coefficient subrings of R. does not exceed X- As we have seen lYI >X, we get lYI =X.
Next we shall consider the uniqueness of coefficient subrings. A finite local ring T is called of type if T is generated by two units a and b such that (1) ab v:ba, (2) a b e J(T), and (3) o(a)=o(b)=lT/J(T)l 1. If T is a finite local ring of type ), then T* is not a nilpotent group. For, let us suppose that T is a finite local ring of type ). Let a and b be generators of T satisfying (1)-(3). Let A and B be cycic subgroups of T* generated by a and b respectively. Let K T/J(T) GF(pr). Then IAI BI p" is coprime to J(T)I. If T* is nilpotent, then A B, as both A and B are complement subgroups of + J(T) in T*. This contradicts (1), so we see that T* is not nilpotent.
THEOREM 3.2. Let R be a local ring with radical M. Assume that M is nilpotent, and K R/M is a commutative field of characteristic p (p a prime) which is algebraic over GF(p). Then the following are equivalent. (i) R has a unique coefficient subring. (ii) R* is a nilpotent group. (iii) R* is isomorphic to the direct product of K* and + M. (iv) R* has no finite local subring of type ).
PROOF. (i) (ii). Clear from Lemma 2.1 (III) and Theorem 2.3 (III). (i) :=:, (iii). Let rt* rt I,. R* K* be the group homomorphism induced by the natural homo- morphism n:R K. Since R has a unique coefficient subring, by Theorem 2.3 (liD, there exists a unique right inverse of n*. Then R* is a semidirect product of + M and K*. Let z be any fixed element of +M. The mapping It:K* R* defined by K* 9 etl- z-k(ot.)z is a right inverse of*, so It by our hypothesis. This implies that each element of %(K*) commutes with each element of + M. Hence R* is the direct product of + M and k(K*). (iii) =::, (iv). Let us suppose that R contains a finite local subring U of type (1). By the proof of 10, Lemma ], + M is a nilpotent group. If R* is isomorphic to the direct product of K* and + M, then R* is nilpotent. So U* is nilpotent, which is a contradiction. (iv) :=> (i). Assume that R has at least two different coefficient subrings. Then there exist at least two different right inverses I and l.t of *. Let {K,}'__, be a sequence of finite subfields of K such that K, K, +, and 7: K, K. There exists a numberj such that I Ix, It Ix,.. Let ],be a generator of K*. Then the subring ((?), t(?)) of R is a finite local ring of type (1). 4. From [9, p. 373, Theorem XIX.4 (b)] and the proofofTheorem 3.1, one may expect that, in Theorem 2.3, any two coefficient subrings of R are always conjugate. However, from the following example, we see that this is incorrect. COEFFICIENT SUBRINGS OF CERTAIN LOCAL RINGS 459
Let K w7=GF(p '), where {r,}7: is a strictly ncreasing sequence of positive integers such that r, r, (i > 1). Let {or,}'= be automorphisms of K such that or, is not the identity on GF(p") (i > 1) and, forj < i, is the on V a or, identity GF(pr). Let 0)7= tKx be left K-vector space with basis {x,}7__ i. We can regard V as a (K,K)-bimodule by defining (E, c,x,)a ,, c, or,(a)x, (Y, c,x, e V,a K). The abelian group R K 9 V together with the multiplication (a, y) (b, z) (ab, az + yb) (a, b K, y, z V) forms a local ring with radical M (0, V), which satisfies the assumption of Theorem 2.3. The homo- morphism rt:R K defined by (a,x) a gives the isomorphism R/M--K. The subring S { (a, 0) a e K} of R is a coefficient subring of R. For each > 1, let y, be a generator of GF(p- Then we can write for a suitable ')*. ?, ?, integer m,. We shall define elements {u,}7__ ofR* inductively as follows: Let u (q( ,x). For u,, (Z,, '__ rjxj) (rj e K), let a, {T,,-%(Y.)}-' {y,,+-%(y.+)}r (1
v -I (b,,0)v (s-,-s-(E, d, x,)s -) (b,, O)(s, Z, d,x,) (b,, Z ,(s-'b, d s-'d%(b,))x), which yields
So, for any > 1, we see d, : 0. This contradicts that ,, d, x, is an element of the direct sum V @7'= ,Kx,.
In conclusion, we shall state a theorem which is a generalization of [3, Theorem].
THEOREM 4.1. (cf [9, p. 376, Theorem XIX.5] and [4, p. 491, Theorem 72.19]) LetR be aring with 1. Assume that J(R) is nilpotent. Let R/J(R)= (K,).,., @ (K2) . @ @ (Ka),,,,,,, where each K,(1 < < d) is a commutative field of characteristic p (p a prime) which is algebraic over GF(p). Then there exists a subring T of R which satisfies the following. 460 T. SUMIYAMA
(i) R T N (as abelian groups), where N s an additive subgroup of R. (ii) T is isomorphic to a finite direct sum of matrix rings over IG-rings. (iii) J(T)=TJ(R)=pT. (iv) T/pT is naturally isomorphic to R/J(R). Moreover,, if T' is another subring of R satisfying (ii)-(iv), then T' is isomorphic to T.
PROOF. Let R=R/J(R)=Re,(R)R-d2@...(R)R-d,, where each R-d,(l
only if k 1. Similarly, (f,, +J;2 + +fi,,,,) + (f2, +f..2 + +f )+ + (fd, +fa'. + +fa.,.), where f, are mutually orthogonal primitive idempotents of T', and T'f, =_ T'fj (as left T'-modules) if and only if k I. As e,Tek,/pek,Te, ek,Re,/e,J(R)e,, we see that e, and ft., are primitive idempotents of R. Then R @Rek, )Rfj are indecomposable decompositions. By what was stated above, Krull-Schmidt theorem tells us that there exists a permutation o" of { 1,2 d} such that n, =mo,)and Re, --Rfo, as left R-modules (I < T'= ,r,),,, ,,, ),,2x,, x,,,' where e, Te, andf T'f are IG-rings. Hence, to complete the proof it will suffice to show e, Te, =_ f T'f. As e,Te, is an IG-ring which is naturally isomorphic to e,Re,/e, J(R)e,, so e,Te, is a coefficient subring of e,Re,. Similarly, fT'f is a coefficient subring of fRf. As e,Re, End(Re,) =_ End(Rf) --fRf, we see e,Te, --fT'f by Theorem 2.3 (II). Note. It is unknown when the subring T of Theorem 4.1 is unique up to inner automorphism of R (see [3, Problem]. ACKNOWLEDGEMENT. The author would like to express his indebtedness and gratitude to Prof. Y. Hirano and Dr. H. Komatsu for their helpful suggestions and valuable comments. REFERENCES [1] BEHRENS, E. A.,Ring Theory, Academic Press (1972). [2] BOURBAKI, N., lgments de Mathgtnatique, AlgSbre, Hermann (1962). [3] CLARK, W. E., A coefficient ring for finite noncommutative rings, Proc. Amer. Math. Soc. 33 (1972), 25-28. [4] CURTIS, C. W. & REINDER, I., Representation Theory of Finite Groups and Associative Algebras, Pure & Appl. Math. Ser. 11, Wiley-Interscience (1962). [5] HASSE, H., Zahlentheorie, 2, Auflage, Akademie-Verlag (Berlin, 1963). [6] JANUSZ, G. T., Separable algebras over commutative rings, Trans. A. M. S. 122 (1966), 461-479. [7] JACOBSON, N., Lectures in Abstract Algebra, vol. III, Van Nostrand (1964). [8] KRULL, W., Algebraische Theorie der Ringe, I, II and III, Math. Annalen $8 (1923), 80-122, 91 (1924), 1-46 and 92 (1924), 183-213. [9] MCDONALD, B. R., Finite Rings with Identity, Pure & Appl. Math. Ser. 28, Marcel Dekker (1974). 10] MOTOSE, K. & TOMINAGA, H., Group rings with nilpotent unit groups, Math. J. Okayama Univ. 14 (1969), 43-46. 11 NAGATA, M., Local Rings, Wiley-Interscience (1962). [12] RAGHAVENDRAN, R., Finite associative rings, Compositio Math. 21 (1969), 195-229. [13] ROBINSON, D. J. S.,A Course in the Theory ofGroups, Springer-Verlag (1982). [14] ROQUETTE, P., Abspaltung des Radikals in vollstfindigen lokalen Ringen, Abh. Math. Sem. Univ. Hamburg 23 (1959), 75-113. [15] SERRE, J. P., Local Fields, Springer-Verlag (1979). 462 T. SUMIYAMA [16] SUMIYAMA, T., On double homothetisms of rings and local rings with finite residue fields, Math. J. Okayama Univ. 33 1991 ), 13-20. [17] WILSON, R. S., Representations of finite rings, Pacific J. Math. 53 (1974), 643-649.